Let’s stand a little back from the Brexit trainwreck – the kind you get when Dr. Evil hacks the signalling at Clapham Junction in rush hour. I have no choice, since as an expatriate I, and a million like me, get no vote.
The options are:
A – Exit with no deal
B – the May deal
C – A softer Brexit (“BINO”) on Norway or Jersey lines; undefined, but probably with staying in most of the EU single market, few restrictions on movement from EU countries, and no say in the rulemaking
D – Remain.
The estimable Simon Wren-Lewis estimates the current factional breakdown of the House of Commons (n=630) over Brexit:
Brexiters – No Deal 100
May loyalists – No freedom of movement 200
People’s Vote [second referendum] 150
Corbyn loyalists 30
Soft Brexit 150
This leads to the following first-choice vote predictions:
A: 100 for, 530 against
B: 200 for, 430 against (actual vote was 202 to 432)
C: <180 for, >450 against
D: <150 for, >480 against
There is a large majority against anything at all. A neater real-life example of the Condorcet paradox you couldn’t get.
Kenneth Arrow wrapped Condorcet in bomb-proof mathematics to show (hope I get this right) that there is no possible voting scheme (in the most general sense of an algorithm that aggregates ranked preferences) obeying very plausible ground rules that offers a guarantee against circular majorities.
One difficulty with the Arrow proof is that it doesn’t suggest how likely the paradox is. If it’s an asteroid strike risk, we can ignore it for practical purposes. As the number of participants and the number of choices both grow, it’s plausible that anomalies get smoothed out and the system gets better behaved.
It would for instance be hard to get a voting paradox over US health care. The policy space can be represented as a continuum between a full tax-funded NHS at one end and an unregulated health and health insurance market at the other. Voters at the poles will have second and third choices reflecting the continuum of options. Democratic voting lands you somewhere in the middle, as is indeed the case. But you can’t be dogmatic about this. I venture an argument here that when policies are in a means-ends relationship, circular majorities are in fact pretty likely.
Brexit is different to either in that the choices are discontinuous and heavily constrained by institutional factors: the Article 50 process, the EU’s negotiating position, the Good Friday Agreement about Northern Ireland. Not to mention outright lunacy on the part of the hard Brexiters.
The way you assemble a majority, in the Commons or the country, from the pieces of the trainwreck is sequential ranked voting. But then it’s crucial who goes first. Imagine a second referendum with the structure:
Q1 = approve policy A, Yes/No
If the majority is against Q1, then
Q2 = approve policy B, Yes/No
If the majority is against Q2, then
Q3 = approve policy C, Yes/No
If the majority is against Q3, then
Q4 = approve policy D, Yes/No
To guarantee a result, you can omit the last question and just say that D is the default.
As I’m a Remainer, I would like the list to start with the crazy hard Brexit and go A→B→C→D. That gives Remain the best chance. But Brexiters would like the reverse order: D→C→B→A. This favours their preferred option. I can’t see a good formal argument for either scheme. What I can point to is the near-universal example of rules of procedure in deliberative assemblies: you vote first on the most radical amendment, then the next most radical. This scheme has a conservative bias towards the status quo, which looks like common sense. We know the status quo works minimally in that it has allowed the assembly to meet and deliberate.